We can see that this is all new to you as you don't have the right names for things!
First thought is that you do not have inverters but generators!
They may have had some reference to inverters in ads and such but if you are wanting to make AC power for the RV, that will be a generator!
An inverter uses battery power to make AC!
When your generators are working, they will power an item called a converter and that turns the AC power from the generators into DC power to work the DC things in the RV like the water pump. The excess 12 volt DC power is also used to charge the batteries!
The lights showing levels will be on a "monitor panel in many cases. But that is different than the thermostat as the thermostat is what you use to control the temperature from heater or air conditioner!
Maybe near each other or on the same panel but two completely different things!
No big thing to find some confusion when first starting and we have all had the same type problems!
But that is all a secondary question if you hear the pump running!
If there is water in the tank and the pump is running but not getting water at faucets, there is a good chance a valve is not closed right!
Does the pump just keep running when turned on but all faucets are closed? It should build up pressure in the lines and then stop until the faucet is opened.
If it runs and doesn't stop, there is likely to be a valve open, letting air into or out of the lines. It's like it is blowing air into a balloon that has a hole in it!
One of the drains might be open but that should let water run out. No water running out under the RV?
Sure there really is water in the tank?
There is often a tube near the water pump to use to stick in a jug of antifreeze to suck it in to keep lines from freezing. Do you know of a clear plastic tube that has one end loose that you could stick in a jug? That should have a way to close the end and maybe a cap on it? There should also be a valve at that tube to switch from sucking antifreeze to regular RV use.
That valve is an easy one to forget to put back to normal use when we get the RV out of storage.
Good luck and just know that we all have to learn lots of things on the new toys!